But as groined vaults are absolutely necessary in ware houses, and many other buildings, for the purpose of giv ing communication throughout the same in every direc tion, which cannot be obtained where waggon-head vaults are used and as there is a great saving of materials, re sulting from the arches bearing on piers only, instead of parallel walls, it follows, that an improvement in their con struction, which tends to bring diem nearer to an equality in strength with the waggon-head vaulting, must be a great acquisition. This improvement has been lately made by Mr George Tapper, an architect of London, who, instead of the square piers hitherto used, has adopted octangular ones, (by which a considerable saving of room is made,) idid has thrown stout tibs diagonally over the vault, whose breadths are equal to the sides of the octagons on which they stand. The side arches, which in brick-wm k are set four inches back from the face of the ribs, in order to save the trouble and expence of cutting the groin angles, are worked into, and rest upon these ribs. By this improved construction, the ribs form a much stronger suppor t for the weight of the incumbent vault, and the loading above, than by the old method; as the following comparison will chew.
Since the force of the superincumbent weight has been found by experience to act chiefly in the direction of the groins, they. require the greatest strength that can be given them ; at the same time, the side arches should be made to throw as little weight as possible upon them, particular ly about the summit of the vault, that their tendency to fracture towards the crown may be diminished as much as the nature of the case will admit. In the groined vaults hi therto constructed, where the groins spring from the ex treme corners of the square piers, their section, taken per pendicular to their length, forms a square, (see a, Plate CCLXXXIV. Fig. 2.) hdving one of its angles turned to wards the centre of the curve, and its side equal to the thickness of the arches ; whereas in the new vault, the sec tion taken in the same manner, forms a rectangle equal in breadth to the side of an octagon inscribed in the square pier, and of a depth which may exceed that of the old groin in any proportion, (see b, Fig. 2.) Now, the strength of an arch, in its different points, is measured by the greatest weight which it is able to carry on those points without breaking ; that is, it is in a ratio compounded of the tripli cate ratio of the secant of the angle of the curve's inclina the horizon, in its vat ious points, and the reciprocal simple ratio of the radius of curvature in the same points. But since the relative strength of arches is to be determin ed by comparing them in their weakest parts, (namely the crown,) the strength of any arch at the crown will be reci procally as the radius of curvature at that point, since the angle of the curve's inclination to the horizon at that point is always 0° 0'; or reciprocally as the span of the arches, when the rise of them is the same, their thickness being supposed equal. From this • it will be evident, that the
strength of the diagonal ribs and groins will be directly as the areas of their sections and distances of the centres of gravity from the place where the fracture would end, and reciprocally as the spans of the arches: that is, if A repre sent the area of the section, G the distance of its centre of gravity, and S the span of the arch ; then the strength of one arch to that of another will be as A . G S Let the side of the square pier be =a ; then the thick ness necessary for the side arches will be -- 4 con sequently, will be the side of the square section of the groins in the old arch ; its area, therefore, will be • and its half diagonal = G. The distance of the ' 8 — piers being 4.5 a, the span of the diagonal groins will be = V2 X (4.5 = 6.364 a=S ; we have therefore A •G 32V 8 • =-- for the strength of the old groin. 6.364 The side ofthe square pier remaining still =a in the new arch, the breadth of the rib —a, which is the side of an octagon inscribed within the vier, the thickness pro a Al per for the rib will be then will A—a —a ; .3 • and S = 5.364 a + ; consequently A G , — , tne strength of the new rib.
122.004 and this will be the case while the same proportions ale preserved, whatever may be the extent of the arch.
If the side arches, with their spandrils, and the loading on the floor above, be cut by parallel vertical planes, (as at S, Fig. 2 ) the sections will be to each other as their chords c d nearly ; which being less in the new vault than in the old, on account of the greater breadth of the ribs, the weight incumbent on those ribs (which always tends to destroy their equilibrium) is just so much less in the former than in the latter, and consequently their tendency to fracture is .diminished in the same proportion.
It may not be improper to observe, that Dr Hutton re commends the stones of the wall, or spandril over the ex trados of the voussoirs of an arch, to be bonded into the stones of the pier, and with one another ; because the pier will then carry part of their weight, and thereby oppose a greater power of resistance to the thrust of the arch. For the same reason, it would be equally advisable, in the new method of building groined vaults, to carry up at the same time the diagonal ribs, side arches, and spandrils, well bonding the whole together into one solid mass ; which will render vaults built in this manner a valuable acquisi tion in warehouses and other large buildings, where the greatest weights are to be sustained.